The element in your photo was a mains element, so I foolishly assumed it was a mains 
fuse that is blowing.
I am not sure what the relay is for, but if it is intended to run off the mains wiring, it won't do anything on a 9v dc supply.
I don't recall there being a mains relay in the Carver Cascade I had, though. 
If it is a 12v 
fuse that's the problem, the mains element and its resistance aren't relevant.
If I recall correctly, there was a control switch with an on button and an off button, which I think was for the gas. I'm not convinced it didnt't just work as a simple switch.
The electric side was indeed a simple on/off switch. This was over 25 years ago, so I may be misremembering.
My recollection was that the cascade was a really clever design let down by the abysmal quality of aluminium casting and pressed steel components. 
Bits corroded really badly and had to be fettled regularly. 
Bits round the burner section was prone to dissolve into reddish-brown dust, but if you cleared thst away, it still worked.
The main hot water tank was a sheet steel drum wrapped in expanded polystyrene insulation with the uninsulated aluminium end caps sealed by rubbery sealing strips. 
If you had hard water in rhe tank, scale used to get under the seals and cause leaks, adding to the corrosion.
If yours was still working, that's quite an achievement.